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Molecular Distance Geometry Problem

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Encyclopedia of Optimization

Article Outline

Introduction

ABBIE Algorithm

Global Continuation Algorithm

D.C. Optimization Algorithms

Geometric Build-up Algorithm

BP Algorithm

Conclusion

Acknowledgements

References

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References

  1. Alfakih AY, Khandani A, Wolkowicz H (1999) Solving Euclidean distance matrix completion problems via semidefinite programming. Comput Optim Appl 12:13–30

    Article  MathSciNet  MATH  Google Scholar 

  2. An LTH (2003) Solving large-scale molecular distance geometry problems by a smoothing technique via the Gaussian transform and d.c. programming. J Global Optim 27:375–397

    Article  MathSciNet  MATH  Google Scholar 

  3. An LTH, Tao PD (2003) Large-scale molecular optimization from distance matrices by a d.c. optimization approach. SIAM J Optim 14:77–114

    Article  MathSciNet  MATH  Google Scholar 

  4. Berger B, Kleinberg J, Leighton T (1999) Reconstructing a three-dimensional model with arbitrary errors. J ACM 46:212–235

    Article  MathSciNet  MATH  Google Scholar 

  5. Berman HM, Westbrook J, Feng Z, Gilliland G, Bhat TN, Weissig H, Shindyalov IN, Bourne PE (2000) The protein data bank. Nucl Acids Res 28:235–242

    Article  Google Scholar 

  6. Blumenthal LM (1953) Theory and Applications of Distance Geometry. Oxford University Press, London

    MATH  Google Scholar 

  7. Brooks III CL, Karplus M, Pettitt BM (1988) Proteins: a theoretical perspective of dynamics, structure, and thermodynamics. Wiley, New York

    Google Scholar 

  8. BrĂ¼nger AT, Nilges M (1993) Computational challenges for macromolecular structure determination by X-ray crystallography and solution NMR-spectroscopy. Q Rev Biophys 26:49–125

    Article  Google Scholar 

  9. Coleman TF, Shalloway D, Wu Z (1993) Isotropic effective energy simulated annealing searches for low energy molecular cluster states. Comput Optim Appl 2:145–170

    Article  MathSciNet  MATH  Google Scholar 

  10. Coleman TF, Shalloway D, Wu Z (1994) A parallel build-up algorithm for global energy minimizations of molecular clusters using effective energy simulated annealing. J Global Optim 4:171–185

    Article  MathSciNet  MATH  Google Scholar 

  11. Creighton TE (1993) Proteins: structures and molecular properties. Freeman and Company, New York

    Google Scholar 

  12. Crippen GM, Havel TF (1988) Distance geometry and molecular conformation. Wiley, New York

    MATH  Google Scholar 

  13. Dattorro J (2005) Convex optimization and euclidean distance geometry. Meboo Publishing USA, Palo Alto

    Google Scholar 

  14. De Leeuw J (1977) Applications of convex analysis to multidimensional scaling. In: Barra JR, Brodeau F, Romier G, van Cutsem B (eds) Recent developments in statistics. North-Holland, Amsterdam, pp 133–145

    Google Scholar 

  15. De Leeuw J (1988) Convergence of the majorization method for multidimensional scaling. J Classif 5:163–180

    Article  MATH  Google Scholar 

  16. Dong Q, Wu Z (2002) A linear-time algorithm for solving the molecular distance geometry problem with exact inter-atomic distances. J Global Optim 22:365–375

    Article  MathSciNet  MATH  Google Scholar 

  17. Dong Q, Wu Z (2003) A geometric build-up algorithm for solving the molecular distance geometry problem with sparse distance data. J Global Optim 26:321–333

    Article  MathSciNet  MATH  Google Scholar 

  18. Eren T, Goldenberg DK, Whiteley W, Yang YR, Morse AS, Anderson BDO, Belhumeur PN (2004) Rigidity, computation, and randomization in network localization. In: Proc IEEE Infocom 2673–2684, Hong Kong

    Google Scholar 

  19. Floudas CA, Pardalos PM (eds)(2000) Optimization in computational chemistry and molecular biology. Nonconvex optimization and its applications, vol 40. Kluwer, The Netherlands

    Google Scholar 

  20. Glunt W, Hayden TL, Hong S, Wells J (1990) An alternating projection algorithm for computing the nearest euclidean distance matrix. SIAM J Matrix Anal Appl 11:589–600

    Article  MathSciNet  MATH  Google Scholar 

  21. Glunt W, Hayden TL, Raydan M (1993) Molecular conformations from distance matrices. J Comput Chem 14:114–120

    Article  Google Scholar 

  22. Glunt W, Hayden TL, Raydan M (1994) Preconditioners for distance matrix algorithms. J Comput Chem 15:227–232

    Article  Google Scholar 

  23. Gunther H (1995) NMR Spectroscopy: basic principles, concepts, and applications in chemistry. Wiley, New York

    Google Scholar 

  24. Havel TF (1991) An evaluation of computational strategies for use in the determination of protein structure from distance geometry constraints obtained by nuclear magnetic resonance. Prog Biophys Mol Biol 56:43–78

    Article  Google Scholar 

  25. Havel TF (1995) Distance geometry. In: Grant DM, Harris RK (eds) Encyclopedia of nuclear magnetic resonance. Wiley, New York, pp 1701–1710

    Google Scholar 

  26. Hendrickson BA (1991) The molecule problem: determining conformation from pairwise distances. Ph.D. thesis. Cornell University, Ithaca

    Google Scholar 

  27. Hendrickson BA (1995) The molecule problem: exploiting structure in global optimization. SIAM J Optim 5:835–857

    Article  MathSciNet  MATH  Google Scholar 

  28. Huang HX, Liang ZA (2003) Pardalos PM Some properties for the euclidean distance matrix and positive semidefinite matrix completion problems. J Global Optim 25:3–21

    Article  MathSciNet  MATH  Google Scholar 

  29. Kearsley AJ, Tapia RA, Trosset MW (1998) The solution of the metric STRESS and SSTRESS problems in multidimensional scaling by Newton's method. Comput Stat 13:369–396

    MATH  Google Scholar 

  30. Kostrowicki J, Piela L (1991) Diffusion equation method of global minimization: performance for standard functions. J Optim Theor Appl 69:269–284

    Article  MathSciNet  MATH  Google Scholar 

  31. Kostrowicki J, Piela L, Cherayil BJ, Scheraga HA (1991) Performance of the diffusion equation method in searches for optimum structures of clusters of Lennard-Jones atoms. J Phys Chem 95:4113–4119

    Article  Google Scholar 

  32. Kostrowicki J, Scheraga HA (1992) Application of the diffusion equation method for global optimization to oligopeptides. J Phys Chem 96:7442–7449

    Article  Google Scholar 

  33. Kuntz ID, Thomason JF, Oshiro CM (1993) Distance geometry. In: Oppenheimer NJ, James TL (eds) Methods in Enzymology, vol 177. Academic Press, New York, pp 159–204

    Google Scholar 

  34. Lavor C, Liberti L, Maculan N (2005) Grover's algorithm applied to the molecular distance geometry problem. In: Proc. of VII Brazilian Congress of Neural Networks, Natal, Brazil

    Google Scholar 

  35. Lavor C, Liberti L, Maculan N (2006) Computational experience with the molecular distance geometry problem. In: Pintér J (ed) Global optimization: scientific and engineering case studies. Springer, New York, pp 213–225

    Google Scholar 

  36. Lavor C (2006) On generating instances for the molecular distance geometry problem. In: Liberti L, Maculan N (eds) Global optimization: from theory to implementation. Springer, Berlin, pp 405–414

    Google Scholar 

  37. Lavor C, Liberti L, Maculan N (2006) The discretizable molecular distance geometry problem. arXiv:q-bio/0608012

    Google Scholar 

  38. Laurent M (1997) Cuts, matrix completions and a graph rigidity. Math Program 79:255–283

    MathSciNet  Google Scholar 

  39. Liberti L, Lavor C, Maculan N (2005) Double VNS for the molecular distance geometry problem. In: Proc. of MECVNS Conference, Puerto de la Cruz, Spain

    Google Scholar 

  40. Man-Cho So A, Ye Y (2007) Theory of semidefinite programming for sensor network localization. Math Program 109:367–384

    Article  MathSciNet  MATH  Google Scholar 

  41. Moré JJ, Wu Z (1996) \( \epsilon \)-Optimal solutions to distance geometry problems via global continuation. In: Pardalos PM, Shalloway D, Xue G (eds) Global minimization of non-convex energy functions: molecular conformation and protein folding. American Mathematical Society, Providence, IR, pp 151–168

    Google Scholar 

  42. Moré JJ, Wu Z (1996) Smoothing techniques for macromolecular global optimization. In: Di Pillo G, Gianessi F (eds) Nonlinear Optimization and Applications. Plenum Press, New York, pp 297–312

    Google Scholar 

  43. Moré JJ, Wu Z (1997) Global continuation for distance geometry problems. SIAM J Optim 7:814–836

    Article  MathSciNet  MATH  Google Scholar 

  44. Moré JJ, Wu Z (1997) Issues in large scale global molecular optimization. In: Biegler L, Coleman T, Conn A, Santosa F (eds) Large scale optimization with applications. Springer, Berlin, pp 99–122

    Google Scholar 

  45. Moré JJ, Wu Z (1999) Distance geometry optimization for protein structures. J Global Optim 15:219–234

    Article  MathSciNet  MATH  Google Scholar 

  46. Neumaier A (1997) Molecular modeling of proteins and mathematical prediction of protein structure. SIAM Rev 39:407–460

    Article  MathSciNet  MATH  Google Scholar 

  47. Palmer KA, Scheraga HA (1992) Standard-geometry chains fitted to X-ray derived structures: validation of the rigid-geometry approximation. II. Systematic searches for short loops in proteins: applications to bovine pancreatic ribonuclease A and human lysozyme. J Comput Chem 13:329–350

    Article  Google Scholar 

  48. Phillips AT, Rosen JB, Walke VH (1996) Molecular structure determination by convex underestimation of local energy minima. In: Pardalos PM, Shalloway D, Xue G (eds) Global minimization of non-convex energy functions: molecular conformation and protein folding. American Mathematical Society, Providence, IR, pp 181–198

    Google Scholar 

  49. Piela L, Kostrowicki J, Scheraga HA (1989) The multiple-minima problem in the conformational analysis of molecules: deformation of the protein energy hypersurface by the diffusion equation method. J Phys Chem 93:3339–3346

    Article  Google Scholar 

  50. Pogorelov A (1987) Geometry. Mir Publishers, Moscow

    Google Scholar 

  51. Saxe JB (1979) Embeddability of weighted graphs in k-space is strongly NP-hard. In: Proc. of 17th Allerton Conference in Communications, Control, and Computing, 480–489, Allerton, USA

    Google Scholar 

  52. Trosset M (1998) Applications of multidimensional scaling to molecular conformation. Comput Sci Stat 29:148–152

    Google Scholar 

  53. Wang L, Mettu RR, Donald BR (2005) An algebraic geometry approach to protein structure determination from NMR data. In: Proc. of the 2005 IEEE Computational Systems Bioinformatics Conference, Stanford, USA

    Google Scholar 

  54. Wu D, Wu Z (2007) An updated geometric build-up algorithm for solving the molecular distance geometry problem with sparse distance data. J Global Optim 37:661–673

    Article  MathSciNet  MATH  Google Scholar 

  55. Wu Z (1996) The effective energy transformation scheme as a special continuation approach to global optimization with application to molecular conformation. SIAM J Optim 6:748–768

    Article  MathSciNet  MATH  Google Scholar 

  56. WĂ¼trich K (1989) The development of nuclear magnetic resonance spectroscopy as a technique for protein structure determination. Acc Chem Res 22:36–44

    Article  Google Scholar 

  57. WĂ¼trich K (1989) Protein structure determination in solution by nuclear magnetic resonance spectroscopy. Science 243:45–50

    Article  Google Scholar 

  58. Zou Z, Byrd RH, Schnabel RB (1997) A stochastic/perturbation global optimization algorithm for distance geometry problems. J Global Optim 11:91–105

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors would like to thank CNPq, FAPESP and FAPERJ for their financial support.

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Lavor, C., Liberti, L., Maculan, N. (2008). Molecular Distance Geometry Problem . In: Floudas, C., Pardalos, P. (eds) Encyclopedia of Optimization. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-74759-0_400

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